The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the following

Proposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.

The problem treated here is an abstract version of the Cauchy problem for an equation of mixed type in the hyperbolic region with initial data on the parabolic line (cf. 2, 3, 5, 11, 13, 14, 15, 16, 21, 27). A more complete bibliography may be found in (3, 5, 18). We begin with the equation (6)

(1.1)

Given a homomorphic mapping θ of a subgroup A of a group G onto another subgroup B of G, necessary and sufficient conditions for the existence of a supergroup G* of G and an endomorphism θ* of G* such that θ* coincides with θ on A were derived by B. H. Neumann and Hanna Neumann (3). The homomorphism θ is called a partial endomorphism of G and θ* is said to continue, or extend, θ. Necessary and sufficient conditions for the simultaneous continuation of two partial endomorphisms of a group G to total endomorphisms of one supergroup G* ⊇ G were derived by the author (2).

As is well known in the theory of graphs a tree is a connected graph without cycles. Many characterizing properties of trees are known (1), for example the cyclomatic number is equal to zero, which is also equal to p — 1, where p is the number of connected components of the graph. The graphs with cyclomatic number equal to p — 1 are defined here as tree-equivalent graphs. A tree is always a tree-equivalent graph but not conversely. The properties of tree-equivalent graphs are studied here.

The theory of non-negative matrices was initiated by Perron (7) and Frobenius (4). Wielandt in (11) gives an elegant exposition of the subject.

It is well known to workers in the field that if a matrix A has all its entries non-negative real numbers, then the pattern of zeros and non-zeros of A completely determines the pattern of zeros and non-zeros in every power of A. Ptak in (8) and Ptak and Sedlacek in (9) describe this behaviour in terms of some combinatorial constructs.

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and define

c(K) = V(K′)/V(K)

Let

c(n) = inf{c(K) : K ⊂ En}.

The purpose of this paper is to prove that the general derivative of a completely additive singular set-function defined on certain measurable subsets of an abstract measure space is zero almost everywhere. As a corollary the celebrated Lebesgue decomposition theorem has been sharpened.

This result is well known for set-functions defined on measurable subsets of an n-dimensional Euclidean space (2, p. 119). The proof in this setting depends on two things: Vitali's covering theorem and the fact that for every measurable set A there exists an open set O which contains A and the images of O and A under the set-function can be made arbitrarily close. Here the covering theorem is due to Trjitzinsky and the open set is replaced by an envelope, an entirely measure-theoretic concept.

An H-space is a topological space T for which it is possible to define a continuous binary composition

with the following properties: there exists a homotopy unit, i.e. an element u ∊ T such that

In (1) Baer introduced the concept of (C, γ)-transitivity and (C, γ)- homogeneity. A projective plane (see (5) for the requisite definitions and axioms) is (C, γ)-transitive if, given an ordered pair (P1, P2) of points collinear with C but distinct from C and not on γ, there is a collineation which maps P1 into P2 and leaves fixed every point on γ as well as every line through C. A projective plane is (C, γ)-homogeneous for a non-incident point-line-pair if it is (C, γ) transitive and there is a correlation which maps every line through C into its intersection with γ and every point on γ into its join with C.

In this paper we obtain a partial answer in graph-theoretic form to a question raised by Ryser (2, p. 68) concerning the minimal number of interchanges required to transform equivalent (0, l)-matrices into each other.

For given positive integers m and n we consider the collection of m × n (0, 1)-matricesA = {aij}, i.e. aij = 0 or 1 for 1 ≤ i ≤ m, 1 ≤ j ≤ n. We say the (0, 1)-matrices A = {aij} and B = {bij} are equivalent and write A ~ B if and only if they have the same row and column sums, that is, if and only if.

Un anneau R est dit primitif (3) s'il existe un R-module irréductible et fidèle M. L'objet de cet article est d'étudier une généralisation naturelle de la notion d'anneau primitif en remplaçant "irréductible" par "sous-directement irréductible" dans la définition ci-dessus. D'une façon plus précise, un anneau R sera dit métaprimitif s'il existe un R-module sous-directement irréductible et fidèle M, avec SR ≠ {0}, S étant le sous-module minimal de M. Nous établissons pour les anneaux métaprimitifs un théorème de structure (théorème 2.3) faisant apparaître certaines analogies avec le théorème de structure correspondant pour les anneaux primitifs.

In a recent paper on isotopy invariants (1), S. T. Hu denned the enveloping space Em(X) of any given topological space X for each integer m > 1. By an application of the Smith theory to the singular cohomology of the enveloping space Em(X), he obtained his immersion classes for every n = 1, 2, 3, . . . and proved (3) the main theorem that a necessary condition for a compact metric space X to be immersible into the ^-dimensional Euclidean space Rn is . This theorem was proved earlier by W. T. Wu (4) for finitely triangulable spaces X using purely combinatorial methods.

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces

In an earlier work, we defined and described a new group algebra , which is a von Neumann algebra containing the group G (3). In this paper we continue this study be relating the lattice of normal subgroups of the group G to the lattice of central projections of the group algebra . More precisely, we shall exhibit a one-to-one mapping ϕ of the lattice of closed normal subgroups of G into the lattice of central projections of , having the property that if N1 ⊂ N2, then ϕ(N2) ≤ ϕ(N1).

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

In this paper we prove the following characterization theorem (Section 3) :

Theorem 1. If each of g and m is a real-valued non-decreasing function on the number interval [a, b], then the following two statements are equivalent: (1) If R is a real-valued, refinement-unbounded (Section 3) function of subintervals of [a, b], then the integral (Section 2)

exists and is equal to g (b) — g (a), and

(2) g is absolutely continuous with respect to m.

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by

(1.1)

where μ(r) is the Möbius function. It follows from (1) that

(12)

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).

Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).